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6. The notion of module
This notebook is part of the Introduction to manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).
If F is a field, a set V is called a vector space over F if there is an operation of addition from V×V to V, and a scalar multiplication operation from F×V to V, such that the following properties are satisfied.
(i) (u+v) + w = u + (v+w) for all u, v, w ∈ V,
(ii) u+v=v+u for all u, v ∈ V,
(iii) There exists an element 0 ∈ V such that 0+v=v+0 for all v ∈ V,
(iv) For each v ∈ V there exists u ∈ V such that v+u=0,
(v) 1v=v for all v ∈ V,
(vi) for all and v∈V,
(vii) for all and v∈V,
(viii) for all and ∈V.
The formal definition of the module M over the ring R is exactly as above (with V replaced by M and F replaced by R) but we relax the requirement that F be a field, and instead allow an arbitrary ring R (with unity). We shall restrict ourselves to the commutative rings R.
A free module of finite rank over a commutative ring R is a module over R that admits a finite basis, that is a finite family , which spans , i.e., for every , for some R, and are linearly independent i.e., implies that all are zero.
Since R is commutative, it has the invariant basis number property, so that the rank (dimension) of the free module M is defined uniquely, as the cardinality of any basis of .
General remark on free modules in SageMath
Basic motivation for introducing free modules into consideration in SageMath Manifolds is the fact that the sets of vector fields and tensor fields on a parallelizable open subset U of the manifold, are free modules over the ring of scalar fields on U.
Some frequently used commands from SageMath FiniteRankFreeModule
:
In our examples we will use mainly the SageMath symbolic ring SR.
Example 6.1
FiniteRankFreeModule
in SageMath
:
Symbolic ring SR is considered as a field:
so the finite rank free modules over SR are in fact vector spaces:
Defining V as a finite rank free module over a field we obtain a vector space without predefined basis.
Later we shall see that without any specific coordinate choice, no basis can be distinguished in a tangent space. This is one more motivation for using modules in SageMath Manifolds.
Let us check that defining vector space with VectorSpace command, we introduce a predefined basis.
In SageMath the VectorSpace of rank 4 over R is in fact R^4 -the Cartesian power of R.
but an analogous FiniteRankFreeModule is not:
Example 6.2
To define a vector in SageMath Manifolds
we need some basis.
Vectors can be defined also as linear combinations of the basis.
Linear transformations in finite rank free module
By the linearity of the map between two modules over we mean the condition
Automorphism
Automorphism of the module M is a bijective linear transformation M M.
An automorphism allows for example for defining a new basis.
If more than one bases are defined, the first one is the default one.
Example 6.3
Let us define a module with two bases and an automorphism defining the change of basis.
Matrix of the automorphism:
Example 6.4
The new basis can be also defined component by component.
Matrix of the change of basis e f:
Matrix of the change of basis f e:
Endomorphism
Endomorphism of the module M is a general linear transformation M M.
Example 6.5
Define an endomorphism using arbitrarily chosen matrix.
Using general symbolic matrices
Example 6.6
To obtain general matrices with elements we can use for example the following definition:
General endomorphism of a 4-dimensional module:
Homomorphisms of modules
The general linear maps between modules M, N are homomorphisms.
A bijective homomorphism is called isomorphism.
Example 6.7
Define two modules and a homomorphism between them.
Let us check the additivity property using general vectors V, W.
The value of the homomorphism phi on the vector V in the basis of N:
The matrix of the homomorphism phi:
Linear forms on modules. Dual module
By a linear form or a linear functional on a module over we mean the map such that
The dual module is the module of linear forms on a module .
Example 6.8
The module of all linear functionals or linear forms on M in SageMath Manifold
is M.dual()
:
Dual basis
SageMath dual.basis
allows to define a dual basis to a given basis , i.e., the set of linear functionals such that
Example 6.9
Define dual basis in SageMath Manifolds
.
Let us check that :
Linear forms on a module M are linear combinations of elements of the dual basis.
Example 6.10
Define a linear form in SageMath Manifolds
.
Linear forms are elements of the dual module .
What's next?
Take a look at the notebook Smooth functions and pullbacks.